Let `f'(sin x) lt 0` and `f''(sin x) gt 0 AA x in (0, pi/2)` and` g (x) = f (sin x) + f (cos x)`, then

Options

A. g' is increasing
B. g' is decreasing
C. g' has a point of minima
D. g' has a point of maxima

Answer

1

Explanation

`g(x)=f(sinx)cosx-f(cosx)sinx`
or `g(x)=f(sinx)sinx+cos^(2)xf(sinx)`
`f(cos x)sin^(2)x-f(cosxgt0forall x in (0,pi//2)`
[as it is given f(sinx)= `f(cos x (pi//2-x))lt0`
Thus g(x) is increasing in `(0,pi//2)`. Also `g(pi//4)=0`
or `g(x)gt0forallx in ((pi)/(4),(pi)/(2))`
and `g(x) ltforall x in (0,pi//4)`
Thus g(X) is decreasing in `(0,pi//4)`